Properties

Label 2.3060.4t3.l.a
Dimension $2$
Group $D_{4}$
Conductor $3060$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(3060\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.45900.4
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.340.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-15}, \sqrt{51})\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 12x^{2} + 51 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 26\cdot 79 + 52\cdot 79^{2} + 31\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 10\cdot 79 + 4\cdot 79^{2} + 54\cdot 79^{3} + 34\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 67 + 68\cdot 79 + 74\cdot 79^{2} + 24\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 70 + 52\cdot 79 + 26\cdot 79^{2} + 47\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,4)$
$(1,2)(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character valueComplex conjugation
$1$$1$$()$$2$
$1$$2$$(1,4)(2,3)$$-2$
$2$$2$$(1,2)(3,4)$$0$
$2$$2$$(1,4)$$0$
$2$$4$$(1,3,4,2)$$0$